Reviews: First order expansion of convex regularized estimators

The present paper proposes an approximation, based on the first order Taylor expansion of convex regularizer. In the regularized regression setting and under some mild condition on the loss function and the underlying distribution that generates the data, the authors prove that one can replace the regularization term of the regression algorithm by its Taylor approximation and have a guarantee that the solution obtain with this approximation will be close to the original solution (according to the Mahalanobis distance). The authors give then examples of such proxy for square loss and logistic regression and also for Constrained Lasso, Penalized Lasso and Group Lasso. The paper also proposes a discussion where this approach can be useful. Although this paper is a bit technical, it is well written and the result are on my opinion non trivial and interesting.